Boundary Value Problems and Sharp Inequalities for Martingale Transforms

2012

Math. Z.

In the paper we study LlogL estimates for Fourier multipliers resulting from modulation of the jumps of Lévy processes. We exhibit a class of functions m :for which the corresponding multipliers T m satisfy the following estimate: for K > 1, any locally integrable function f on R d and any Borel subsetwhere (t) = (t + 1) log(t + 1) − t. We also present related lower bounds which arise from considering appropriate examples for the Beurling-Ahlfors operator.

Section: A Martingale Inequalitymentioning

confidence: 99%

Logarithmic inequalities for Fourier multipliers

2012

Math. Z.

“…So far, the best results in this direction is the inequality ||B|| L p (C)→L p (C) ≤ 1.575(p * − 1) of Bañuelos and Janakiraman, and the estimate ||B|| L p (C)→L p (C) ≤ 1.4(p − 1) if p ≥ 1000, due to Borichev, Janakiraman and Volberg [8]. Both these bounds were established with the use of probabilistic methods; more precisely, the proofs rest on certain martingale inequalities of Burkholder [9], [10] (see also Wang [20]) and their appropriate extensions.…”

Section: Introductionmentioning

confidence: 99%

Sharp Logarithmic Bounds for Beurling-Ahlfors Operator Restricted to the Class of Radial Functions

2013

Mediterr. J. Math.

Abstract. We establish a class of sharp logarithmic estimates for the Beurling-Ahlfors transform B on the complex plane. For anywhere Ψ(t) = (t + 1) log(t + 1) − t and B(0, R) ⊂ C denotes the ball of center 0 and radius R. A related result in higher dimensions is also established. The proof rests on probabilistic methods and exploits a certain sharp inequality for martingales. Mathematics Subject Classification (2010). Primary 42B20; Secondary 60G44.

“…Such a type of problems appears in many places in the literature, in the study of other classical operators and objects in harmonic analysis. See e.g., Melas [12], Melas et al [13] and Slavin et al [19] for related problems concerning the dyadic maximal operators; consult Burkholder [3] for related results for martingale transforms and the Haar system on [0, 1]; Vasyunin [22] studied similar questions for A p -weights on the real line; Slavin and Vasyunin [20] investigated similar problems for BMO functions on R; and more. Coming back to (1.3) and the restriction (1.4), we easily see that if Y = |X|, then the lower bound for ||F || L p (T) can be improved.…”

Section: Introductionmentioning

confidence: 99%

Sharp L p Bound for Holomorphic Functions on the Unit Disc

2014

Mediterr. J. Math.

Abstract. For any 1 < p < ∞ and any X, Y ∈ R satisfying |X| ≤ Y , we determine the optimal constant Cp(X, Y ) such that the following holds. If F is a holomorphic function on the unit disc satisfying ReThis can be regarded as a reverse version of the classical estimates of Riesz and Essén. The proof rests on the exploitation of certain families of special subharmonic functions on the plane.Mathematics Subject Classification. Primary 31B05; Secondary 30J99.